A025035 -id:A025035 - cp's OEIS frontend

A060542 a(n) = (1/6)multinomial(3n;n,n,n).

1, 15, 280, 5775, 126126, 2858856, 66512160, 1577585295, 37978905250, 925166131890, 22754499243840, 564121960420200, 14079683012144400, 353428777651788000, 8915829964229105280, 225890910734335847055, 5744976449471863238250, 146603287914300510042750

Offset: 1

Henry Bottomley, Apr 02 2001

Number of ways of dividing 3n labeled items into 3 unlabeled boxes with n items in each box.
From Antonio Campello (campello(AT)ime.unicamp.br), Nov 11 2009: (Start)
A060542(t) is the number of optimal [n,2,d] binary codes that correct at most t errors, i.e., having Hamming distance 2*t + 1 (achieved on length n = 3*t + 2). These codes are all isometric.
It is also the number of optimal [n,2,d] binary codes that detect 2*t + 1 errors, i.e., having Hamming distance 2t+2 (obtained by adding an overall parity check to the n = 3*t + 2 optimal codes). These codes are also all isometric.
For t = 0, we have the famous MDS, cyclic, simplex code {(000), (101), (110), (011)}. (End)
Also the number of distinct adjacency matrices of the complete tripartite graph K_{n,n,n}. - Eric W. Weisstein, Apr 21 2017

Harry J. Smith, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Adjacency Matrix
Eric Weisstein's World of Mathematics, Complete Tripartite Graph

Row 3 of A060540.

Cf. A006480, A025035, A361717.

a:= n-> combinat[multinomial](3*n,n$3)/3!:
seq(a(n), n=1..18);  # Alois P. Heinz, Jul 29 2023

Table[(3*n)!/(n!^3*6),{n,1,20}] (* Vaclav Kotesovec, Sep 23 2013 *)
Table[Multinomial[n, n, n], {n, 20}]/6 (* Eric W. Weisstein, Apr 21 2017 *)

{ a=1/6; for (n=1, 100, write("b060542.txt", n, " ", a=a*3*(3*n - 1)*(3*n - 2)/n^2); ) } \\ Harry J. Smith, Jul 06 2009

a(n) = (3*n)!/((n!)^3*6) = a(n-1)*3*(3*n - 1)*(3*n - 2)/n^2 = A060540(3,n) = A006480(n)/6. - corrected by Vaclav Kotesovec, Sep 23 2013
a(n) ~ 3^(3*n-1/2)/(4*Pi*n). - Vaclav Kotesovec, Sep 23 2013
a(n) = 1/(8*n^3) * Sum_{k = 0..2*n} (-1)^(n+k) * k*(2*n-k)^2 * binomial(2*n, k)^3. - Peter Bala, Oct 11 2024
a(n) = 1/(n^2) * Sum_{k = 0..n} (-1)^(n+k+1) * (n-k)^2 * binomial(2*n, k)^3. - Peter Bala, Nov 03 2024

Definition revised by N. J. A. Sloane, Feb 02 2009

A157703 G.f.s of the z^p coefficients of the polynomials in the GF2 denominators of A156925.

Original entry on oeis.org

1, 1, 5, 5, 2, 62, 152, 62, 2, 91, 1652, 5957, 5957, 1652, 91, 52, 5240, 77630, 342188, 551180, 342188, 77630, 5240, 52, 12, 8549, 424921, 5629615, 28123559, 61108544, 61108544, 28123559, 5629615, 424921, 8549, 12

Offset: 0

Johannes W. Meijer, Mar 07 2009

The formula for the PDGF2(z;n) polynomials in the GF2 denominators of A156925 can be found below.
The general structure of the GFKT2(z;p) that generate the z^p coefficients of the PDGF2(z; n) polynomials can also be found below. The KT2(z;p) polynomials in the numerators of the GFKT2(z;p) have a nice symmetrical structure.
The sequence of the number of terms of the first few KT2(z;p) polynomials is: 1, 1, 2, 5, 6, 9, 12, 13, 16, 19, 22, 23, 26. The first differences follow a simple pattern. The positions of the 1's follow the Lazy Caterer's sequence A000124 with one exception, here a(0) = 0.
A Maple algorithm that generates relevant GFKT2(z;p) information can be found below.

			Some PDGF2 (z;n) are:
  PDGF2(z;n=3) = (1-z)^3*(1-2*z)^2*(1-3*z)
  PDGF2(z;n=4) = (1-z)^4*(1-2*z)^3*(1-3*z)^2*(1-4*z)
The first few GFKT2's are:
  GFKT2(z;p=0) = 1/(1-z)
  GFKT2(z;p=1) = -z/(z-1)^4
  GFKT2(z;p=2) = z^2*(5+5*z)/(1-z)^7
Some KT2(z,p) polynomials are:
  KT2(z;p=2) = 5+5*z
  KT2(z;p=3) = 2+62*z+152*z^2+62*z^3+2*z^4
  KT2(z;p=4) = 91+1652*z+5957*z^2+5957*z^3+1652*z^4+91*z^5

Originator sequence A156925.
See A000292 for the z^1 coefficients and A040977 for the z^2 coefficients divided by 5.
Row sums equal A025035.
Cf. A157702, A157704, A157705.

p:=2; fn:=sum((-1)^(n1+1)*binomial(3*p+1,n1) *a(n-n1),n1=1..3*p+1): fk:=rsolve(a(n) = fn,a(k)): for n2 from 0 to 3*p+1 do fz(n2):=product((1-m*z)^(n2+1-m),m=1..n2): a(n2):= coeff(fz(n2),z,p): end do: b:=n-> a(n): seq(b(n), n=0..3*p+1); a(n)=fn; a(k)=sort(simplify(fk)); GFKT2(p):=sum((fk)*z^k,k=0..infinity); q2:=ldegree((numer(GFKT2(p)))): KT2(p):=sort((-1)^p*simplify((GFKT2(p)*(1-z)^(3*p+1))/z^q2),z, ascending);

PDGF2(z;n) = Product_{m=1..n} (1-m*z)^(n+1-m) with n = 1, 2, 3, ...
GFKT2(z;p) = (-1)^(p)*(z^q2)*KT2(z, p)/(1-z)^(3*p+1) with p = 0, 1, 2, ...
The recurrence relation for the z^p coefficients a(n) is a(n) = Sum_{k=1..3*p+1} (-1)^(k+1)*binomial(3*p + 1, k)*a(n-k) with p = 0, 1, 2, ... .

A323293 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have two vertices in common.

Original entry on oeis.org

1, 1, 1, 2, 5, 26, 271, 5596, 231577, 21286940, 4392750641, 2100400533176

Offset: 0

Gus Wiseman, Jan 10 2019

			The a(5) = 26 hypergraphs:
  {}
  {{1,2,3}}
  {{1,2,4}}
  {{1,2,5}}
  {{1,3,4}}
  {{1,3,5}}
  {{1,4,5}}
  {{2,3,4}}
  {{2,3,5}}
  {{2,4,5}}
  {{3,4,5}}
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
Non-isomorphic representatives of the 6 unlabeled 3-uniform hypertrees spanning 6 vertices where no two edges have two vertices in common, and their multiplicities in the labeled case which add up to a(6) = 271:
    1 X {}
   20 X {{1,2,3}}
   90 X {{1,2,5},{3,4,5}}
   10 X {{1,2,3},{4,5,6}}
  120 X {{1,3,5},{2,3,6},{4,5,6}}
   30 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}

Essentially the same as A287232.

Cf. A000665, A025035, A125791, A190865, A289837, A302374, A302394, A319540, A320395, A322451, A323292-A323299.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]>1&]],{n,8}]

a(9) from Andrew Howroyd, Aug 14 2019

a(10) and a(11) (using A287232) from Joerg Arndt, Oct 12 2023

A334056 Triangle read by rows: T(n,k) is the number of configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 3n.

Original entry on oeis.org

1, 0, 1, 7, 2, 1, 219, 53, 7, 1, 12861, 2296, 226, 16, 1, 1215794, 171785, 13080, 710, 30, 1, 169509845, 19796274, 1228655, 53740, 1835, 50, 1, 32774737463, 3260279603, 170725639, 6250755, 178325, 4137, 77, 1, 8400108766161, 727564783392, 32944247308, 1036855344, 25359670, 507584, 8428, 112, 1

Offset: 0

Donovan Young, Apr 15 2020

In this generalized game of memory n indistinguishable triples of matched cards are placed on the vertices of the path of length 3n. A polyomino is a triple on three adjacent vertices. For dominoes in ordinary memory on the path of length 2n, see A079267.

T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k of the sets being a contiguous set of elements. - Andrew Howroyd, Apr 16 2020

			The first few rows of T(n,k) are:
      1;
      0,    1;
      7,    2,   1;
    219,   53,   7,  1;
  12861, 2296, 226, 16, 1;
  ...
For n=2 and k=1 the polyomino must start either on the second vertex of the path, or the third, otherwise the remaining triple will also form a polyomino; thus T(2,1) = 2.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020. See also J. Int. Seq., Vol. 23 (2020), Article 20.9.1.

Row sums are A025035.

Cf. A079267, A334057, A334058, A325753.

CoefficientList[Normal[Series[Sum[y^j*(3*j)!/6^j/j!/(1+y*(1-z))^(3*j+1),{j,0,20}],{y,0,20}]],{y,z}]

T(n,k)={sum(j=0, n-k, (-1)^(n-j-k)*(n+2*j)!/(6^j*j!*(n-j-k)!*k!))} \\ Andrew Howroyd, Apr 16 2020

G.f.: Sum_{j>=0} (3*j)! * y^j / (j! * 6^j * (1+(1-z)*y)^(3*j+1)).

T(n,k) = Sum_{j=0..n-k} (-1)^(n-j-k)*(n+2*j)!/(6^j*j!*(n-j-k)!*k!). - Andrew Howroyd, Apr 16 2020

A110100 a(n) is the number of 2-regular 3-hypergraphs on 3n labeled vertices. (In a 3-hypergraph, each hyper-edge is a proper 3-set; 2-regular implies that each vertex is in exactly 2 hyperedges.)

Original entry on oeis.org

1, 0, 75, 122220, 757275750, 12713292692100, 474415445827323000, 34461884930947363890000, 4431555785100983345799993000, 939388724430508823324694340500000

Offset: 0

Marni Mishna, Jul 11 2005

P-recursive

			One of the 75 2-regular 3-hypergraphs on 6 vertices: {1,2,3} {4,5,6} {1,2,4} {3,5,6}.

Denis S. Krotov, Konstantin V. Vorob'ev, On unbalanced Boolean functions attaining the bound 2n/3-1 on the correlation immunity, arXiv:1812.02166 [math.CO], 2018.
Marni Mishna, Maple worksheet

Cf. A025035, A110101, A110103, A001205.

Recurrence: {a(0) = 1, a(1) = 0, (361631520*n + 1358261784*n^2 + 2841968052*n^3 + 3241507005*n^5 + 3725654130*n^4 + 1922779782*n^6 + 781684101*n^7 + 214347870*n^8 + 37889775*n^9 + 3897234*n^10 + 177147*n^11 + 39916800)*a(n) + (870112800*n + 1655958600*n^2 + 1805971896*n^3 + 561697416*n^5 + 1244162430*n^4 + 166255740*n^6 + 31125384*n^7 + 3346110*n^8 + 157464*n^9 + 199584000)*a(n + 1) + (70976400*n + 86362056*n^2 + 57212568*n^3 + 5161320*n^5 + 22352760*n^4 + 653184*n^6 + 34992*n^7 + 24393600)*a(n + 2) + (-468192*n-411840-198432*n^2-37152*n^3-2592*n^4)*a(n + 3) + 64*a(n + 4), a(2) = 75, a(3) = 122220}.
Differential equation satisfied by generating series A(t)=sum a(n) t^(3n)/(3n)!: {F(0) = 1, 16*t^5*(-2 + t^3)^3*(d^2/dt^2)F(t) + 8*t*(t^9-20*t^3 + 8)*(-2 + t^3)^2*(d/dt)F(t) + t^6*(t^3 + 10)*(t^3-4)*(-2 + t^3)^2*F(t)}.
a(n) ~ 3^(4*n+1/2) * n^(4*n) / (2^n * exp(4*n+1)). - Vaclav Kotesovec, Mar 11 2014

Replaced broken link, Vaclav Kotesovec, Mar 11 2014

A110101 a(n) is the number of 3-regular 3-hypergraphs on n labeled vertices. (In a 3-hypergraph, each hyper-edge is a proper 3-set; 3-regular implies that each vertex is in exactly 3 hyperedges.)

Original entry on oeis.org

1, 0, 0, 0, 1, 12, 330, 11205, 505505, 28787052, 2024844444, 172592502570, 17545270969545, 2098273032696720, 291739927315433454, 46676360010342811203

Offset: 0

Marni Mishna, Jul 11 2005

P-recursive

			The 3-regular 3-hypergraphs on 4 vertices: {1,2,3}, {2,3,4},{3,4,1},{4,1,2}.

Marni Mishna, Maple worksheet

Cf. A025035, A110100, A002829.

Differential equation satisfied by exponential generating function: {F(0) = 1, 36*t^2*(t + 1)*(t^2 - 2)^2*(3*t^2 + 2*t - 2)^2*(d^2/dt^2)F(t) - 12*(t + 1)*(3*t^2 + 2*t - 2)*(3*t^10 + 2*t^9 - 8*t^8 - 40*t^7 - 56*t^6 + 4*t^5 - 48*t^4 - 96*t^3 + 80*t^2 + 80*t - 32)*(d/dt)F(t) + t^3*(t + 1)*(3*t^2 + 2*t - 2)*(3*t^9 + 2*t^8 - 2*t^7 - 108*t^6 - 144*t^5 + 32*t^4 - 24*t^3 + 16*t^2 + 112*t - 64)*F(t)}.
Linear recurrence for a(n): initial values: a(2) = 0, a(3) = 0, a(0) = 1, a(1) = 0, a(4) = 1, a(5) = 12, a(6) = 330, a(7) = 11205, a(8) = 505505, a(9) = 28787052, a(10) = 2024844444, a(11) = 172592502570, a(12) = 17545270969545;
then (1971620508*n^4 + 4242044664*n^3 + 3*n^12 + 4459328640*n + 1437004800 +
167310*n^9 + 5794678656*n^2 + 20779902*n^7 + 234*n^11 + 8151*n^10 + 2248389*n^8
+ 618210450*n^5 + 134970693*n^6)*a(n) + (154*n^10 + 77519860*n^5 + 334620440*n^4
+ 958003200 + 5280*n^9 + 106260*n^8 + 1392666*n^7 + 12460602*n^6 + 979793232*n^3
+ 1848236544*n^2 + 2014882560*n + 2*n^11)*a(n + 1) + ( - 96300*n^7 - 1200066*n^6
- 540148032*n^2 - 767940480*n - 4980*n^8 - 57398920*n^4 - 219822600*n^3
- 479001600 - 10060470*n^5 - 2*n^10 - 150*n^9)*a(n + 2) + ( - 97416*n^8
- 17244057600 - 24771847680*n - 2808*n^9 - 36*n^10 - 1978992*n^7 - 26064612*n^6
- 232501752*n^5 - 1422206064*n^4 - 5889271968*n^3 - 15795689472*n^2)*a(n
+ 3) + ( - 5364230400*n - 4790016000 - 24*n^9 - 1872*n^8 - 64368*n^7 - 1280160*n^6
- 16223256*n^5 - 135808848*n^4 - 750702432*n^3 - 2641118400*n^2)*a(n + 4)
+ (3252704*n^5 + 2043740160 + 194208*n^6 + 2058817536*n + 33702144*n^4 +
221164160*n^3 + 897495552*n^2 + 6560*n^7 + 96*n^8)*a(n + 5) + (246432*n^6
+ 48931572*n^4 + 4055546880 + 1512709248*n^2 + 4406952*n^5 + 7824*n^7 +
345350856*n^3 + 108*n^8 + 3758813568*n)*a(n + 6) + (528439296*n + 2696360*n^4
+ 27036368*n^3 + 161115712*n^2 + 159784*n^5 + 5208*n^6 + 72*n^7 + 735989760)*a(n
+ 7) + ( - 59595808*n^2 - 8517816*n^3 - 338532480 - 504*n^6 - 680168*n^4
- 220837728*n - 28776*n^5)*a(n + 8) + ( - 262432*n^3 - 288*n^5 - 11355392*n
- 13824*n^4 - 20613120 - 2459328*n^2)*a(n + 9) + (31392*n^3 + 3713184*n
+ 720*n^4 + 512496*n^2 + 10074240)*a(n + 10) + (253440 + 288*n^3 + 8544*n^2
+ 82176*n)*a(n + 11) + ( - 7584*n - 49536 - 288*n^2)*a(n + 12) + 384*a(n + 13).
a(n) ~ n^(2*n) * 3^(n+1/2) / (exp(2*n+2) * 4^n). - Vaclav Kotesovec, Mar 11 2014
Recurrence (of order 11): 192*(243*n^2 - 285*n - 290)*a(n) = 144*(n-1)*(243*n^3 - 285*n^2 + 34*n + 796)*a(n-1) + 48*(n-2)*(n-1)*(1701*n^2 - 24*n + 1027)*a(n-2) - 48*(n-3)*(n-2)*(n-1)*(729*n^3 - 2556*n^2 - 2601*n + 2558)*a(n-3) + 8*(n-3)*(n-2)*(n-1)*(3645*n^3 - 7110*n^2 - 35091*n + 30676)*a(n-4) + 12*(n-4)*(n-3)*(n-2)*(n-1)*(729*n^4 - 4500*n^3 + 1623*n^2 + 12924*n - 11872)*a(n-5) + 8*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(729*n^3 - 4338*n^2 - 1728*n + 3269)*a(n-6) - 8*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(486*n^2 + 2145*n - 3485)*a(n-7) - 12*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(243*n^3 - 1014*n^2 - 1304*n + 1619)*a(n-8) + 24*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(18*n - 13)*a(n-9) - 6*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(27*n - 71)*a(n-10) + (n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(243*n^2 + 201*n - 332)*a(n-11). - Vaclav Kotesovec, Mar 11 2014

Replaced broken link, Vaclav Kotesovec, Mar 11 2014

A323296 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have exactly one vertex in common.

Original entry on oeis.org

1, 0, 0, 1, 11, 10, 25, 406, 4823, 15436, 72915, 895180, 11320441, 71777498, 519354927, 6155284240, 82292879425, 788821735656, 7772567489083, 98329764933354, 1400924444610675, 17424772471470490, 216091776292721021, 3035845122991962688, 46700545575567202903

Offset: 0

Gus Wiseman, Jan 11 2019

nonn

The only way to meet the requirements is to cover the vertices with zero or more disconnected 3-uniform hypergraphs with each edge having exactly two vertices in common (A323294). - Andrew Howroyd, Aug 18 2019

			The a(4) = 11:
  {{1,2,3},{1,2,4}}
  {{1,2,3},{1,3,4}}
  {{1,2,3},{2,3,4}}
  {{1,2,4},{1,3,4}}
  {{1,2,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
The following are non-isomorphic representatives of the 3 unlabeled 3-uniform hypergraphs spanning 7 vertices with no two edges having exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 406.
  210 X {{1,2,3},{4,6,7},{5,6,7}}
  140 X {{1,2,3},{4,5,7},{4,6,7},{5,6,7}}
   21 X {{1,6,7},{2,6,7},{3,6,7},{4,6,7},{5,6,7}}
   35 X {{1,2,3},{4,5,6},{4,5,7},{4,6,7},{5,6,7}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323294, A323297.

b:= n-> `if`(n<5, (n-2)*(2*n^2-6*n+3)/6, n/2)*(n-1):
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, k-1)*b(k)*a(n-k), k=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 18 2019

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]==1&],Union@@#==Range[n]&]],{n,8}]

seq(n)={Vec(serlaplace(exp(-x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x + O(x^(n-1)))/2)))} \\ Andrew Howroyd, Aug 18 2019

From Andrew Howroyd, Aug 18 2019: (Start)
Exponential transform of A323294.
E.g.f.: exp(-x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2). (End)

a(11) from Alois P. Heinz, Aug 12 2019

Terms a(12) and beyond from Andrew Howroyd, Aug 18 2019

A323299 Number of 3-uniform hypergraphs on n labeled vertices where every two edges have exactly one vertex in common.

Original entry on oeis.org

1, 1, 1, 2, 5, 26, 261, 3216, 19617, 80860, 262651, 737716, 1920821, 5013152, 14277485, 47610876, 186355041, 820625616, 3869589607, 19039193980, 96332399701, 499138921736, 2639262062801, 14234781051932, 78188865206145, 437305612997376, 2487692697142251

Offset: 0

Gus Wiseman, Jan 11 2019

nonn

			The a(5) = 26 hypergraphs:
  {}
  {{1,2,3}}
  {{1,2,4}}
  {{1,2,5}}
  {{1,3,4}}
  {{1,3,5}}
  {{1,4,5}}
  {{2,3,4}}
  {{2,3,5}}
  {{2,4,5}}
  {{3,4,5}}
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
The following are non-isomorphic representatives of the 10 unlabeled 3-uniform hypergraphs on 7 vertices where every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 3216.
    1 X {}
   35 X {{1,2,3}}
  315 X {{1,2,5},{3,4,5}}
  105 X {{1,2,7},{3,4,7},{5,6,7}}
  840 X {{1,3,5},{2,3,6},{4,5,6}}
  840 X {{1,4,5},{2,4,6},{3,4,7},{5,6,7}}
  210 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}
  630 X {{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
  210 X {{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
   30 X {{1,2,7},{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}

Cf. A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323296, A323297, A323298.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]!=1&]],{n,8}]

Binomial transform of A323298.

Terms a(11) and beyond from Andrew Howroyd, Aug 14 2019

A124503 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} (or of any n-set) containing k blocks of size 3 (0<=k<=floor(n/3)).

Original entry on oeis.org

1, 1, 2, 4, 1, 11, 4, 32, 20, 113, 80, 10, 422, 385, 70, 1788, 1792, 560, 8015, 9492, 3360, 280, 39435, 50640, 23100, 2800, 204910, 295020, 147840, 30800, 1144377, 1763300, 1044120, 246400, 15400, 6722107, 11278410, 7241520, 2202200, 200200, 41877722

Offset: 0

Emeric Deutsch, Nov 14 2006

Row n contains 1+floor(n/3) terms. Row sums yield the Bell numbers (A000110). T(n,0)=A124504(n). Sum(k*T(n,k), k=0..floor(n/3))=A105480(n+1).

			T(4,1)=4 because we have 1|234, 134|2, 124|3 and 123|4.
Triangle starts:
    1;
    1;
    2;
    4,   1;
   11,   4;
   32,  20;
  113,  80, 10;
  422, 385, 70;
  ...

Alois P. Heinz, Rows n = 0..250, flattened

Cf. A000110, A124504, A105480, A355144.

T(3n,n) gives A025035.

G:=exp(exp(z)-1+(t-1)*z^3/6): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,k),k=0..floor(n/3)) od; # yields sequence in triangular form
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(i=3, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Mar 08 2015

nn = 8; k = 3; Range[0, nn]! CoefficientList[Series[Exp[Exp[x] - 1 + (y - 1) x^k/k!], {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 26 2012 *)

E.g.f.: G(t,z) = exp(exp(z)-1+(t-1)z^3/6).

A200472 Triangle T(n,k) is the number of ways to assign n people to k unlabeled groups of equal size.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 3, 0, 1, 1, 0, 0, 0, 1, 1, 10, 15, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 35, 0, 105, 0, 0, 0, 1, 1, 0, 280, 0, 0, 0, 0, 0, 1, 1, 126, 0, 0, 945, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 462, 5775, 15400, 0, 10395, 0, 0, 0, 0, 0, 1

Offset: 1

Dennis P. Walsh, Nov 18 2011

If k is not a factor of n, T(n,k) = 0. If k is a factor of n, T(n,k) = (n!/k!)/(n/k)!^k. If n is a multiple of k, we may obtain T(n,k) by arranging all n people in an ordered line, which can be done in n! ways. Peel off the first n/k people for "group 1", the next n/k people for "group 2", ..., and the last n/k people for "group k". Since the k groups are actually unlabeled, we must divide n! by k! Also, since the ordering of the n/k people within each of the k groups is not of importance, we must now divide by (n/k)!^k. Therefore, T(n,k) = (n!/k!)/(n/k)!^k.

Also, T(2n,n) provide the sequence consisting of the products of consecutive odd integers.

			Triangle T(n,k) begins
1;
1,   1;
1,   0,    1;
1,   3,    0,     1;
1,   0,    0,     0,   1;
1,  10,   15,     0,   0,     1;
1,   0,    0,     0,   0,     0,  1;
1,  35,    0,   105,   0,     0,  0,  1;
1,   0,  280,     0,   0,     0,  0,  0,  1;
1, 126,    0,     0, 945,     0,  0,  0,  0,  1;
1,   0,    0,     0,   0,     0,  0,  0,  0,  0,  1;
1, 462, 5775, 15400,   0, 10395,  0,  0,  0,  0,  0,  1;
...
T(6,2) = 10 since there are 10 ways to assign 6 people (A,B,C,D,E,F) into 2 groups of size 3. The assignments are {A,B,C}|{D,E,F}, {A,B,D}|{C,E,F}, {A,B,E}|{C,D,F}, {A,B,F}|{C,D,E}, {A,C,D}|{B,E,F}, {A,C,E}|{B,D,F}, {A,C,F}|{B,D,E}, {B,C,D}|{A,E,F}, {B,C,E}|{A,D,F}, and {B,C,F}|{A,D,E}.

Dennis P. Walsh, Note on assigning n people into k unlabeled groups of equal size

T(2n,n) is A001147(n).
T(3n,n) is A025035(n).
T(4n,n) is A025036(n).
Row sums of T(n,k) provide A038041(n).
A200473 is A200472 with zeros removed.

T:= (n, k)-> `if`(modp(n, k)=0, n!/(k!*((n/k)!)^k), 0):
seq(seq(T(n, k), k=1..n), n=1..20);

nn=11;s=Sum[Exp[y x^i/i!]-1,{i,1,nn}];Range[0,nn]!CoefficientList[Series[s,{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Sep 15 2012 *)

T(n,k) = if(n%k!=0, 0, (n!/k!)/((n/k)!)^k );
for (n=1,15, for (k=1,n, print1(T(n,k),", "));print());
/* Joerg Arndt, Sep 16 2012 */

For k that divide n, T(n,k) = (n!/k!)/((n/k)!)^k; otherwise, T(n,k) = 0.
E.g.f. when k is fixed: (1/k!) sum(j>=1, (x^j/j!)^k ).
E.g.f. for T(n*r,n): exp(x^r/r!).
T(2n,n) = (2n-1)!! = (2n-1)(2n-3)...(3)(1).

cp's OEIS Frontend

A060542 a(n) = (1/6)*multinomial(3*n;n,n,n).

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A157703 G.f.s of the z^p coefficients of the polynomials in the GF2 denominators of A156925.

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A323293 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have two vertices in common.

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A334056 Triangle read by rows: T(n,k) is the number of configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 3n.

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A110100 a(n) is the number of 2-regular 3-hypergraphs on 3n labeled vertices. (In a 3-hypergraph, each hyper-edge is a proper 3-set; 2-regular implies that each vertex is in exactly 2 hyperedges.)

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A110101 a(n) is the number of 3-regular 3-hypergraphs on n labeled vertices. (In a 3-hypergraph, each hyper-edge is a proper 3-set; 3-regular implies that each vertex is in exactly 3 hyperedges.)

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A323296 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have exactly one vertex in common.

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A323299 Number of 3-uniform hypergraphs on n labeled vertices where every two edges have exactly one vertex in common.

A060542 a(n) = (1/6)multinomial(3n;n,n,n).